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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2013, Volume 52, Number 1, Pages 57–63 (Mi al571)

Recognizability of alternating groups by spectrum

I. B. Gorshkov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Abstract: The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. It is proved that simple alternating groups An are recognizable by spectrum, for $n\ne6,10$. This implies that every finite group whose spectrum coincides with that of a finite non-Abelian simple group has at most one non-Abelian composition factor.

Keywords: finite group, simple group, alternating group, spectrum of group, recognizability by spectrum.

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English version:
Algebra and Logic, 2013, 52:1, 41–45

Bibliographic databases:

UDC: 512.542
Revised: 04.12.2012

Citation: I. B. Gorshkov, “Recognizability of alternating groups by spectrum”, Algebra Logika, 52:1 (2013), 57–63; Algebra and Logic, 52:1 (2013), 41–45

Citation in format AMSBIB
\Bibitem{Gor13} \by I.~B.~Gorshkov \paper Recognizability of alternating groups by spectrum \jour Algebra Logika \yr 2013 \vol 52 \issue 1 \pages 57--63 \mathnet{http://mi.mathnet.ru/al571} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3113477} \zmath{https://zbmath.org/?q=an:06189478} \transl \jour Algebra and Logic \yr 2013 \vol 52 \issue 1 \pages 41--45 \crossref{https://doi.org/10.1007/s10469-013-9217-x} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000319133000004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884985950} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. B. Gorshkov, “Recognizability of symmetric groups by spectrum”, Algebra and Logic, 53:6 (2015), 450–457
2. E. Jabara, D. Lytkina, A. Mamontov, “Recognizing $M_{10}$ by spectrum in the class of all groups”, Int. J. Algebr. Comput., 24:2 (2014), 113–119
3. Yu. V. Lytkin, “Groups critical with respect to the spectra of alternating and sporadic groups”, Siberian Math. J., 56:1 (2015), 101–106
4. N. V. Maslova, “Finite simple groups that are not spectrum critical”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 211–215
5. N. V. Maslova, “Finite groups with arithmetic restrictions on maximal subgroups”, Algebra and Logic, 54:1 (2015), 65–69
6. M. A. Grechkoseeva, A. V. Vasil'ev, “On the structure of finite groups isospectral to finite simple groups”, J. Group Theory, 18:5 (2015), 741–759
7. A. V. Vasil'ev, “On finite groups isospectral to simple classical groups”, J. Algebra, 423 (2015), 318–374
8. A. M. Staroletov, “On recognition of alternating groups by prime graph”, Sib. elektron. matem. izv., 14 (2017), 994–1010
9. Yu. V. Lytkin, “On finite groups isospectral to $U_3(3)$”, Siberian Math. J., 58:4 (2017), 633–643
10. Yu. V. Lytkin, “On finite groups isospectral to the simple groups $S_4(q)$”, Sib. elektron. matem. izv., 15 (2018), 570–584
11. M. A. Grechkoseeva, “On spectra of almost simple extensions of even-dimensional orthogonal groups”, Siberian Math. J., 59:4 (2018), 623–640
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